Problem: Multiply the following complex numbers, marked as blue dots on the graph: $(3 e^{5\pi i / 4}) \cdot (3 e^{5\pi i / 6})$ (Your current answer will be plotted in orange.)
Answer: Multiplying complex numbers in polar forms can be done by multiplying the lengths and adding the angles. The first number ( $3 e^{5\pi i / 4}$ ) has angle $\frac{5}{4}\pi$ and radius $3$ The second number ( $3 e^{5\pi i / 6}$ ) has angle $\frac{5}{6}\pi$ and radius $3$ The radius of the result will be $3 \cdot 3$ , which is $9$ The sum of the angles is $\frac{5}{4}\pi + \frac{5}{6}\pi = \frac{25}{12}\pi$ The angle $\frac{25}{12}\pi$ is more than $2 \pi$ . A complex number goes a full circle if its angle is increased by $2 \pi$ , so it goes back to itself. Because of that, angles of complex numbers are convenient to keep between $0$ and $2 \pi$ $\frac{25}{12}\pi - 2 \pi = \frac{1}{12}\pi$ The radius of the result is $9$ and the angle of the result is $\frac{1}{12}\pi$.